I can confirm this. Been happening to me on actually any page using the steam browser, and it only occurs after a few hours of usage.
Happens on the current stable release, and it started with the previous stable release.

[1028/172830:FATAL:compositor_software_output_device.cc(83)] Check failed: mem.
[1028/172831:ERROR:renderer_main.cc(227)] Running without renderer sandbox
[1028/172831:WARNING:shared_memory_posix.cc(386)] Shared memory creation failed; out of file descriptors
[1028/172831:ERROR:host_shared_bitmap_manager.cc(122)] Cannot create shared memory buffer
[1028/172831:WARNING:shared_memory_posix.cc(386)] Shared memory creation failed; out of file descriptors
[1028/172831:FATAL:compositor_software_output_device.cc(83)] Check failed: mem.
Exactly the same messages
[1028/172831:ERROR:tcp_socket_libevent.cc(181)] CreatePlatformSocket() returned an error: Too many open files

Any page will look similar to: http://pix.toile-libre.org/upload/original/1414514101.png
Killing all instances of steamwebhelper fixes the issue temporarily until it inevitably happens again

With the latest update, killing the steamwebhelper processes no long fixes the issue. Instead, it renders Steam completely unusable, leaving you no choice but to also kill the steam process itself and log back in

[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.hOnmsz failed: Too many open files
[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.iJ0W9E failed: Too many open files
[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.HOJDRK failed: Too many open files
[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.8YpqzQ failed: Too many open files
[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.Yg0ihW failed: Too many open files
[1113/151154:ERROR:shared_memory_posix.cc(231)] Creating shared memory in /dev/shm/.org.chromium.Chromium.bIchZ1 failed: Too many open files
Shows up in stdout right after Steam starts acting up.

Also, by just killing the steamwebhelper process which contains the following commandline argument, Steam didn't crash and the browser ones again became usable:
http://pix.toile-libre.org/upload/original/1415888248.png

The latest beta (built 11/11) is affected as well

I think this is fixed