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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 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...

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 poison needs to be potentially harmful to people.
  
  Some spiders inject poisons that are very harmful to insects but have little impact on humans. But other spiders inject poisons that are potentially harmful to all animal life.


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

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 poisonous 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.