Heinlein's "all you zombies" is an early version of this.
In terms of biology, it's extremely rare for humans to have fully functioning sets of both reproductive systems. The reproductive physiology develops from the same parts so it either develops "male", "female", or (more common than people think) somewhere in between but not "both". A clone may or may not even develop in the same way.
One other possibility, though also very rare. is near-identical twins. If twins have the same genetics (in other words, they're identical twins) but have intersex conditions that result in different reproductive development it's conceivable that they could reproduce with each other. It's uncommon to be fertile incongruous with your genetics, though, so again: this would be extremely unlikely.
That said, in a sci-fi context there are some options. Biological hermaphroditism is possible in animals, though there aren't any vertebrates on Earth, so conceivably an alien species could encounter this problem.
Advanced medicine could also build people a new reproductive system and implant it, giving them the counterpart they lacked.
Of course, in that case, advanced medicine may also be able to give you someone else's reproductive genetics (ex, implanted sperm or eggs) and thus effectively sidestep the genetic problems. I'm imagining a particularly narcissistic person getting a gender-flipped clone of themself and using someone else's genetic material to have a kid with themself but also avoid the shallow gene pool problem.
Of course, you don't need to go that far. If you're female (or have those parts anyway) you can just do IVF. Don't even need to clone yourself, then.
Anyway, this is getting a little silly.
You can pretty much do this with Firefox. I'm on my phone but ublock origin can block 3rd party cookies and scripts. This breaks a lot of sites but it also lets you turn those back on, on a case-by-case basis. Plus various other Firefox settings.