Can I cash out my German state pension as a lump sum?

Yes, as a refund of your contributions. On request, Deutsche Rentenversicherung pays you a one-off refund of the share you paid yourself, which for employees is 9.3% of your gross pay subject to contributions; it keeps your employer's share (§ 210(3) SGB VI). If you hold no EU, EEA or Swiss passport, you usually get the money once you live outside those countries and 24 calendar months have passed since your last compulsory contribution. With a passport from a treaty state such as the US or India, it usually only works with fewer than 60 contribution months. You can't leave individual years out: the refund covers all contributions together, and afterwards those years no longer count towards any pension (§ 210(6) SGB VI). If you live in Germany, you usually get nothing back before the standard retirement age.