Rust absolutely works on 32-bit platforms, though we often use the i686 target rather than an i386 one. Platform support list is here: https://forge.rust-lang.org/platform-support.html
Theo is talking about building rustc, not compiling most Rust programs. That's the first distinction that it seems like many people are missing.
However, apparently the compilation process OOMs when building on a i386 box. I don't use those platforms, but I'd believe it. The Rust compiler is large. However, I thought (and looking at our CI, this seems to be true https://travis-ci.org/rust-lang/rust/jobs/311223817) we do compile with an i686-unknown-linux-gnu host (for this build), so I dunno. Maybe it was a fluke, maybe I'm misunderstanding, I'm not sure.
We often provide artifacts via cross-compiling, but this is unacceptable to OpenBSD. That's totally okay. They have good reasons for doing this.