I used to be in the same situation as Robert Maxwell
, minus the ATM fees. The bank did have branches and you could call them. I once had to send in my debit card because it had been blocked. But apart from that, I didn't miss the branches.
I'm now with the online branch of a bank that also has real branches - though they're far and between. I only needed their help once. So, in summary, you should be fine but I'd look for one that offers help via phone (with real people on the other end), too, in case something unforeseen should happen. Some internet research should get you some reviews.