With all due respect to my friends Alan and Alan above, according to my TIG instructor and my own personal experience, TIG welding stainless is relatively easy compared to mild steel, and far easier than aluminum. Note I said "relatively".
TIG is by far the best welding method for this use as compared to wire feed, stick or gas. There may be better methods beyond the financial means of small shop welding, but I don't know about them. Disclaimer: I haven't welded pressure/vacuum vessels. (but I suspect it's in my near future...)
(Edit: OK I have, but that was rough, overbuilt and explosion proof, not precision work)
The key (as always) is cleanliness. TIG is inherently clean but never use the same cleaning tools (wire brushes, wipes etc) with any other materials. Keep Acetone handy and keep your filler rod pristine. Never let anything that will touch the weld area touch anything else after cleaning.