I have seen people at the gym who weigh 300lbs and more... on the whole i think most people at gyms are encouraging as they see you are making positive efforts to change your life, and some of them will have been where you are at right now so know what it's like, although there are lots of fit people at the gym, they all had to start somewhere

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Keep an eye on your local newspaper and/or lookup a gym near you and see if they have any 'free' taster passes, my local gym do a 2 day free pass so you can get an idea of if you will like it or not and get a measure of what the average gym-goer there is like, friend or foe, hehe.
As far as only doing short periods of time goes - that's OK too, you are paying them, not them paying you to be there ;-). If you can only do 10 minutes then that's just fine, and with most gyms you pay monthly anyway so can go as frequently as you like, maybe do 10 or 20 minutes a day instead of a big long 1 hour session each week. At the beginning stages you will find your stamina grows very fast if you just keep plugging away at it.