I've been thinking about encryption this morning (and I have no clue why) and I started thinking about if it's at all possible for two strangers to establish a secure connection. I'm having my doubts, but I don't know a whole lot about encryption.

Yes. There are several well-known algorithms that can be used to establish an encrypted communication channel without the cleartext exchange of secret keys. A classic example is the use of Diffie-Hellman, in which a large number is agreed upon (exchanged in the clear) and then each party chooses a relatively prime factor and exchange some information using those factors to discover a mutually common secret key without actually revealing their secrets. Very cunning.

Establishing an encrypted channel between two parties is a relatively simple and straightforward solved problem.

Where you might run into trouble is in mutual authentication. This is an entirely different problem from encrypted communication, although many authentication solutions rely on the establishment of encrypted communication to be effective. Almost all authentication mechanisms rely on a trusted third party somewhere along the line.

Security is a feeling, not a science. Pretty much anything useful has some sort of security vulnerability, at least potentially.

Where you might run into trouble is in mutual authentication. This is an entirely different problem from encrypted communication, although many authentication solutions rely on the establishment of encrypted communication to be effective. Almost all authentication mechanisms rely on a trusted third party somewhere along the line.

But the underlying problem is that encrypted communication is absolutely useless without mutual authentication - and integrity - because you have no guarantee that what you send will be received intact (or at all) by the other party and you don't know (and cannot detect) if someone is impersonating the party you are communicating with. So you really have no privacy nor integrity, in the technical sense, without authentication. DH does not "establish an encrypted communication channel" at all, it permits secure key exchange on an insecure channel if and only if the two parties can authenticate each other. Without this condition it does not provide any security at all and you cannot communicate securely on said insecure channel.

In the theoretical sense, anyway, but an MITM is really not that difficult to mount if you really wanted to.