I weigh around 17 stone = 238 pounds.
Would a 250cc - 350cc or even a 450cc hold my weight?

Yamaha quads.

 

Yes, but you will be most comfortable on the 450 or 700 models. Height is a big factor as well, the 250 is for smaller riders. I am 5'9" and the 250 raptor an blaster models are ok, for about 30 minutes. Then I crave something larger. The utility and 4x4 models tend to run a bit bigger. Try before you buy.

 

Probably a Raptor 660 or 700 would be better. The yfz 450 is great for racing but they require a lot more maintence and need to be revved high to get the power out of them. The bigger engine in the Raptor makes a lot of power but it can just as easily putt around at slower speeds as well.

 

Weigh is not the problem its room. ATVs come in 3 chassis sized small, mid and full.
Example in Honda utility.

Small chassis =250 Recon
Mid chassis= 350/420 Rancher
Full chassis= 500 Foreman/Rubicon

Weight wise even a 250 Recon is rated for much more weight then 250lbs. but do you want your knees up beside your head all day?