More than 160,000 people were trying to access Currys PC World website at one point on Wednesday, which had only a limited supply.
The recommended retail price starts at £449 - but eBay bids are currently running to thousands of pounds.
Experts said a rise in bots buying consoles added to the high demand had led to technical difficulties.
PlayStation said the demand was unprecedented.
On Wednesday morning, a message on the Currys PC World site said it "now has limited PS5 stock".
But it sold out in under an hour.
