The server still wasn't very responsive. I eventually determined that it wasn't CPU, the network wasn't looking odd, and the RAID array was just dandy. A quick check online revealed that access_log_today was probably corrupt. That would explain it—Apache tries to insert a record into that table every time someone accesses a page. I killed the Apache processes (they wouldn't shut down properly via init script), and the hard drive light immediately stopped flashing. I then repaired the table and deleted the records in it for good measure. When I restarted Apache it ran just fine.
Is there anything I can do to ensure that this sort of thing doesn't slow the machine down so much that it takes several minutes to execute a command? I'm using MySQL 4.0.22, Apache 2.0.5something, and Linux 2.6.something.