Build a low budget computer meant for 24/7 operation.

Sometimes you need a machine you can trust to be on for many days at a time. But you're not looking to invest into a server grade build, nor do you need exceptional performance. Let's say it's supposed to run a script or a process, constantly. For example - a DVR server.
In this case you need to optimize your configuration for durability. So to start - we need a nice ventilated chassis. If budget is not unlimited, then a basic well ventilated chassis will do.

Motherboard - We need the motherboard as cheap and as durable for the price as possible. So a good idea is to look through motherboard vendor lineups - go to the cheapest model and then raise the bar till you reach a model with all solid capacitors. For example - currently (end of 2014) Asus has the H81 series motherboards, and one of the cheapest is H81M-K. It however does not have all solid capacitors, so if we take the next model, right above this one - it will be H81M-A - which does offer an all solid capacitor configuration, and is only slightly more expensive.

CPU - if you're not looking for high performance, I would suggest a simple Intel CPU, such as currently Haswell based Pentium Dual core series. They are cheap and bear the Intel quality standard.
You can also go with a T series Intel CPU which consumes the least power.

RAM - again something that is of acceptable quality but not gamer grade. A built in heat-sink is always a plus. As an example - Corsair CMV4GX3M1A1600C11

Hard Drive - This is where you need to invest a bit more into a drive made for 24/7 operation, such as Western Digital Caviar Red made for RAID, NAS and 24 hour operation. Similar models from other manufacturers include Seagate Surveillance HDD or Toshiba High Durability Specialty drives.

PSU - A great way to provide stable power to a long running machine is to go with a good power supply unit that you can trust. Some even come with a built in surge protector. Also make sure you get a PSU with an 80Plus rating, which means it wastes less power. A good example of an affordable 80Plus PSU is Corsair VS350