Probably a difficulty starting the Mathematica Kernel (the part that performs the calculations).

1) Restart the computer and launch Mathematica from a terminal window with the -clean option,This will make sure all preference and other start-up files are reset to default contents.

2) If that does not fix it, does the Mathematica Kernel run by itself?  The Kernel can be launched by entering "math" in a terminal window.
The bare Kernel has a plain-text interface, so you can just type 2+2 ENTER, for example, to verify that it works.
If there are error messages, send them to support@wolfram.com with a description of the computer and OS level.
Copying and pasting the whole Kernel session into the email is usually convenient.

Look for what the system-call utility on your computer is called. ("strace" is common.)

Start the Mathematica Kernel with command line syntax like

   strace -o /tmp/mcrashlog.log   math

substituting the correct name of the trace utility and option.

After a second of hanging, please kill the kernel process and send the mcrashlog.log to Technical Support.
If it is over one MByte, please upload it via http://upload.wolfram.com/, rather than emailing.
For the "Recipient:" field, put   support@wolfram.com  . In the "Additional notes:" field, put
"Community 197049 for Bruce".

Which version of Arch Linux?  What CPU chip?

Thanks everybody for the help! The problem was indeed with my loopback network interface, as Ilian pointed out. I needed to add a rule in my iptables configuration to allow loopback traffic. Now Mathematica is up and running.