The binary for epsxe_x64 is closed-source.  pcsxr will not compile
properly on a modern compiler.  pcsx-redux won't build and is over 1G in
size for just the source.  What's somebody who wants to play PS1 games
on Linux to do?

Well, you're in luck.  Download epsxe205linux_x64.zip from the official
epsxe site.  You'll note that it's linked against some pretty old
libraries if you try to run it.  This is doable, just requires some
fiddling around in /usr/lib64 and compiling an old SSL version.

ln -s /usr/lib64/libncurses.so /usr/lib64/libncurses.so.5
ln -s /usr/lib64/libtinfo.so /usr/lib64/libtinfo.so.5

...the binary doesn't actually use ncurses or tinfo since it's a GTK+
application.  That, however, was the easy part:

Download https://github.com/openssl/openssl/releases/tag/OpenSSL_1_0_2u
.  Yes, you need a 1.0 release.  Extract it.  Its configuration process
is a little wonkier than modern standards usually use.  You need to
"./Configure shared linux-x86_64" ; "./Configure linux-x86_64 shared"
followed by make will get you some very confusing error messages as the
"shared" has to be first and the Configure script does not make that
very clear.

Anyway, after Configure has been run correctly, it should build
the libssl.so.1.0.0 .  However, "make install" will install the
libraries into /usr/local/lib/ssl instead of /usr/local/lib .  You'll
have to move the shared libraries out of /usr/local/lib/ssl/lib and into
/usr/local/lib , then rerun ldconfig.  After this is done, "ldd
epsxe_x64 | grep found" should return nothing, and epsxe_x64 should
start up!  (You're looking for any lines matching "not found" here.)  

NEXT STEPS:

openssl_1_0_2u will eventually stop compiling properly.  I have a prebuilt
libssl.so.1.0.0 and a README on https://crow202.com/2026/libssl-1.0.2.tar.gz ,
so try that if you get stuck on compiling it from source.