Stephen - an upgrade is possible. My own site started in the 3.x days and has been upgraded over time to the current version.
What will be key is:
- doing the upgrade in major version steps (ie, 4.x, 5.x, 6.x, 7.x). Others may say this isn't necessary but I think it's a safe but slow way of going. If you upgrade to each major version then you can check the site each step of the way and verify any fixes.
- if you have third-party code you need to determine the status of the parts for compatibility and have any new versions ready.
- make sure you take backups of files& database after each step, clearly label them and work out as you go.
You should do all of this as a 'dry run' on your local machine or test server if possible, then replicate the steps on your live site.
Good luck and stick with it, the result will be worth the pain.