I don't quite have an answer, but it seems that this must be
something cryptographic, like a password manager, or a Trusted Platform Module, or a PGP keyring, or a full-disc encryption system.
My wee tiny courtyard, a grove of some trees,
I strongly suspect there's a pun here that I will kick myself when I finally see.
entering is easy, but leaving's no breeze.
Encrypting something with someone's public key is easy but decrypting the message again is supposed to be infeasible. OR: Entering your fingerprints into your iPhone's "Secure Enclave" is easy but getting the data out is supposed to be terribly hard. Etc.
Once your innermost enters, I'll hold them quite dear
but after they enter, they never leave in the clear.
Same ideas again. Note that "cleartext" is cryptospeak for the original unencrypted information.
Something else strange, about my fine lair,
never does one go alone, but only in pairs.
Seems like a reference to public-key encryption; or perhaps more elementarily to (username,password) pairs or something of the kind.
Well that's not quite true, only one goes within.
The one hiding outside, in a way, is his twin.
Public and private keys. Or maybe username and password, or something.
One last thing to know, one thing that's quite nice,
You can't force your way in, not for any price.
Strong encryption schemes are supposed to be infeasible to break without more computing hardware than will fit in the universe. Likewise for hardware that tries to embody such schemes.
This is a thing about me, that many admire.
Wrong words wreck all, as if gone in a fire.
Some cryptographic systems (e.g., ones that provide "full disc encryption") are set up to self-destruct after a few wrong attempts at decrypting. Less dramatically, even without any self-destruction, if you attempt to decrypt something with the wrong key, you get garbage essentially indistinguishable from random numbers.
So all these things kinda fit ... but I haven't been able to identify an actual concrete answer that satisfies me. If I have to pick one particular thing, I will reluctantly go for
the "Secure Enclave" in an Apple iPhone
because it's the nearest I have so far found to the "grove of some trees".