VMware ESXi Version 5.1.0
Take a look to the last vpxd.log (check date because they are numbered) and paste the last 200 lines or so.
Since youre running an old vCenter on windows most likely a SQL Express DB is used as the backend. The DB size is limited when using SQL Express and when the vCenter getting older and older and depending on the number of events/tasks you reach the max. DB size which prevents the VXPD(vCenter) to start.
Normaly you see this in the logs as whell when using SQL Studio and connecting to the DB and check the size.
Regards,
Joerg