Here's an important lesson. A lower second class honours degree (or a third for that matter) is not a bar to a good career but it is a bar to your first good job. Once you find another way in to your first good job and build some experience, no one will care what your degree classification is but the challenge is how to get that first good job. What you need to understand is that graduate recruitment is very different to 'normal' recruitment. 'Normal' recruitment relies on work experience and demonstrated success; graduate recruiters can seldom rely on this and instead have to rely on more arbitrary information such as exam results. Graduate recruitment thus tends to be more arbitrary than experienced recruitment as a result. There's a theory that extra curriculars put things right but to be honest I think less than perfect results are often auto-filtered out before the CVs are even read to be honest.
So, what to do? The civil service is certainly one good option. They rely far more heavily on your overall CV, interviews and assessment than they do on degree classification. However, the civil service fast track is very competitive for everyone as a result so you'll need some other options. If you've done law, you might want to give Irwin Mitchell a try. They say in their graduate recruitment that they look beyond academic results; they're worth a go I'd think.
Failing that, you probably need to get into the mindset of a school leaver (with A-Levels) when looking for your first job. Your degree won't have been a waste of time (you'll find it easier to enter professional examinations in a range of fields without doing entry level exams first) but also won't be a ticket into a graduate scheme. In this mindset, you start at the bottom and work your way up. Options might include a bookkeeping role where you can study AAT and eventually do ACCA, CIMA or the ACA. Or maybe you could get a paralegal role and do CILEx, which opens the door to eventually qualifying as a solicitor.
The options are definitely out there but they will involve a lot of work and maybe a little longer at the bottom rung. But five years in, experience becomes king and no one cares about your degree.