In an episode of Peppa Pig, one of the characters says that spiders are very very small, and that they can’t hurt you.

Broadcasters in Australia have avoided broadcasting the episode involved, because they’re worried that viewers will regard the claim as factually true.

Are spiders unable to hurt people? Also, does size have anything to do with whether they can harm you?

I’m primarily interested in whether spiders are harmful to humans, not whether they can affect pigs, and I’m interested in whether it’s true in all countries, though if it’s untrue worldwide, I’d like to know if it is at least true in the country Peppa Pig was created in (the UK), as opposed to the deathworld which is Australia.

The answer is that spiders definitely can hurt you. While you may not be likely to be killed, spiders can absolutely hurt you, whether from a large one's bite (whether venomous or not) or from any venomous spider.

Australia is probably the best case here, and while they have only had one death from spider bite in 40 years (from a redback bite) this is considered to be mostly because:

an effective antivenom for redback spiders was developed in 1956, and one for funnel-web spiders in 1980.

from australianmuseum.net.au

Also on that page:

...on current evidence the most dangerous spiders in the world are funnel-web spiders (Atrax and Hadronyche species), Redback Spiders and their relations (Latrodectus species), Banana Spiders (Phoneutria species) and Recluse Spiders (Loxosceles species). In Australia, only male Sydney Funnel Web Spiders and Redback Spiders have caused human deaths, but none have occurred since antivenoms were made available in 1981.

In the UK (from sciencefocus.com):

three common spiders that are capable of biting you: the cellar spider, the woodlouse spider and the false widow spider. Their bites are painful and have been known to cause swelling for a few hours. In 2014 a 60-year-old woman died after being bitten by a false widow spider.

The false widow, cellar, woodlouse and redback spiders are all pretty small, at around 1cm. A funnel-web can be up to about 5cm, so that probably doesn't count as small

Anecdotally, my son was bitten by a false widow in the summer of 2018 in Scotland, and after falling very ill and having his entire forearm swell up, with the central bite area becoming necrotic, he required emergency treatment, was in hospital for surgery, and a few days stay in the plastic surgery ward at St. Johns Hospital in Livingston. He still has a rather nasty scar on his arm, but thus far has not developed any strange powers...although he certainly sees a lot more spiders now, so maybe that's heightened alertness, maybe he attracts them...

The claim

First let's start with noting that the goal of the episode was to get children over their irrational fear of spiders and making those children see the bigger picture.

Mummy says cobwebs mean spiders and she hates spiders but Daddy Pig doesn't because spiders eat flies and flies are horrid.

Source: Synopsis of episode https://peppapig.fandom.com/wiki/Spider_Web

And taking the claim "spiders are “very very small” and “can’t hurt you”" as a completely literal statement when it's from a show meant for children teaching vague general rules is not correct.

If we for example take a look at animal deaths in the US from 2008 to 2015 we find a lot more deaths due for example dogs than venomous spiders. Now imagine the same episode teaching children to not be afraid of dogs and whether a statement along the lines of "dogs are man's best friend" would be taken literally. It's of course not a 1:1 comparison at all, but the point is just to question whether the statement should be taken completely literally.

Source: wormsandgermsblog.com

Arachnophobia is the irrational fear of spiders (e.g. when my cousin literally was too afraid to pass under a bridge because there might be a spider hiding as there were some (old) cobwebs visible). This irrational fear is more common in children and adolescents and a claim like this needs to be understood in the context of trying to prevent and help people away from such phobias. Point is once again that the claim should be understood as an educational statement trying to help with a social problem.

People with phobia cause discomfort to others, and their behavior is socially unacceptable. Although individuals with phobia are aware that their fear is irrational,they cannot control their feelings. Therefore, a phobia is unquestionably a social dilemma

[...]

The initial age for animal phobia is usually 12 years old.One of the largest epidemiological studies in this field (n =43093) on specific fears was done in the United States and found that the highest prevalence of phobia was in children and adolescents.

Source: Entomophobia and Arachnophobia Among School-Age Children: A Psychological Approach

Important: This is not to say that this excuses such a claim as mild non-phobic fear/aversion of spiders you don't recognize is probably a good idea.

A literal interpretation of the claim

If we do take the claim literally than of course the table above does show that in the US there are just 6 fatalities per year due to venomous spiders ("just" compared to the danger of for example hornets, wasps and bees).

Looking at the home market of the show - the UK - the UK the claim however is even literally pretty reasonable

How many UK spiders are actually dangerous?

Essentially, none.

There are three common spiders in the UK that are capable of biting you: the cellar spider, the woodlouse spider and the false widow spider. Their bites are painful and have been known to cause swelling for a few hours.

In 2014 a 60-year-old woman died after being bitten by a false widow spider. However, the cause of death was due to a bacterial infection from the puncture wound, rather than the spider venom itself.

Source: https://www.sciencefocus.com/nature/how-many-uk-spiders-are-actually-dangerous/

That doesn't mean nobody has an allergic reaction to a spider bites (or otherwise ends up with complications) or that spider bites never cause an infection (just like any scratch) which is why the anecdote in the highest voted answer is so misleading. It's the bigger picture (how common is something bad compared to how afraid people are) which explains why certain fears need to be addressed. In my personal opinion the show could have taken an even more nuanced tone (they suggest leaving them outside the house and not killing them which is already pretty reasonable).

Are spiders unable to hurt people?

There are several species of spiders, some large, others not so large. that are quite capable of harming people. Some can cause severe injury to or kill people. Three requirements:

• The fangs need to be large enough to puncture the epidermis.
  
  The human epidermis is thick enough to render what otherwise would be harmful small creatures impotent with regard to humans.


• The injected venom needs to be potentially harmful to people.
  
  Some spiders inject venoms that are very harmful to insects but have little impact on humans. But other spiders inject venoms that are potentially harmful to all animal life.


• The amount of venom injected needs to be potentially harmful to people.
  
  A spider that injects a tiny amount of what would deadly venom in much larger amounts is not harmful. The dosage makes the venom venomous.

Some spiders pass all three requirements. People have died or lost appendages due to spider bites. Australia appears to be the worst place, but very venomous spiders do exist elsewhere. The Americas, for example, are home to several species of spiders that can be quite harmful to people, and even on occasion can kill. From Spider: This Bites: Venomous Texas Spiders,

The venom of the black widow is a neurotoxin and can lead to severe systemic reactions and in rare cases, death. The black widow's venom is reportedly 15 times more toxic than the venom of the prairie rattlesnake. However, only a minute quantity is injected with each bite.

and

Brown recluse spiders are golden brown in color and can be identified by the characteristic dark brown to black fiddle-shaped pattern on the head region. Brown recluse venom has necrotizing enzymes that generally cause local or systemic reactions.

Black widows are largish (but not huge) spiders, brown recluses, not so large.