I've managed to get into oil & gas, not as an engineer in the traditional sense but doing completions work for O&G projects. I'm part systems manager, part developer for the software that powers the completions side of the project.
As people have said it does seem to be who you know, I fell into this when I was looking for a job last time around, a friend knew a guy who was looking for some support with the software side of things and I ended up being sent out to Texas for 6 months running the system.
A lot of the engineers are old blokes, 50-60 mostly, one guy is 74! Also the money that engineers are on is obscene compared to almost anything else I know of bar a private doctor or dentist, with long hours 70+ a week 7 days for 6 weeks at a time but some guys are on £120 p/h
I do know that some of the guys get work with agents but they won't take new guys without experience. Most work is found through the old boy network, who you worked on other jobs with etc.
To get started you would need find a company who is accepting staff and do a few years on jobs at a normal rate of pay before you could leave and go contract like most of the lads here are.
There can be a lot of travel, but there are also jobs onShore, still plenty of plants need operators etc. I know a process op, again good money (35K for a 25 y/o with a HND) but he does shifts 6 days on 4 off I *think*. I don't get to see him that often but he does enjoy his time off.
If you are young, single and manage to get started early I think it could be a great career but its tough if you have kids and are married, I certainly wasn't expecting to be away this long and its hard on family / friends. For me the money doesn't really make it worth it.