Ken is right but is being a little polite in his use of language here. You don't really have a choice, check the release notes of v2.7. The jump from v2.0 to v2.7 is just too great a leap for the database to handle. The Moodle database is subject to refinement and inclusions and other changes, so the longer the time between versions, the greater the variations in the database.
The database of v2.0 is very different to the database of v2.3, so you are going to have to download v2.2.11 from the Legacy Releases download page and put it on a localhost somewhere, your own laptop or an older otherwise disused PC, restore the v2.0 files, then back them up again, then you have a much better chance of successfully restoring in v2.7. I know, a serious pain, but I am afraid there is no real choice here. Easy to do, and an effective problem solver.
I would seriously recommend that you instigate a regular update of your site, once a year is usually about right on either a large production site, or a small site, you can suit yourself - but not to the latest and greatest, to the most recently superseded. For example, this year, I am upgrading my sites from a v2.5.x to likely a v2.7.2+, not to the v2.8 to be released in November. There are some reasons to go to v2.8, but I prefer things to not be too likely to have something that might bite my clients, so taking one step back has proved a successful update strategy. I have not had any real problems with my sites and the 2.x.x+ has some improvements, but has any problematical coding fixed, any disturbance by plugins repaired.