I could, although if you just want a basic setup it's far easier to just get a HP Microserver when the cashback offer is on. It's not the cheapest setup ever, but it's pretty damn good. Should cost you roughly £200 (without disks) after cashback for the Celeron setup, although you'll likely want to upgrade the memory for another £20-30.
Alternately the following should be better for streaming, having a quad core CPU. It won't be quite as easy on the power consumption as the microserver, but it should manage at least 2 HD streams concurrently (the microserver tends to struggle with more than 1 at a time. It's about the same price as the Microserver if the cashback deal isn't on, at £300 (without disks).
Note: this has been thrown together in about 90 seconds, so I can't promise it's the best thought out setup or that I've not missed anything! I also have no idea whether that case takes a full size ATX PSU
You'd need to add at least a couple of disks (£80 for 1TB mirrored), and I'd probably add a 30-60GB SSD for the OS, if you can justify one on your budget
YOUR BASKET
1 x
Intel Core i5-4440 3.10GHz (Haswell) Socket LGA1150 Processor - Retail £152.99
1 x
Antec ISK600M mATX Cube Case - Black £56.99
1 x
Kingston HyperX Fury Black 8GB (2x4GB) DDR3 PC3-12800C10 1600MHz Dual Channel Kit (HX316C10FBK2/8) £37.99
1 x
MSI H81M-E33 Intel H81 (Socket 1150) DDR3 Micro ATX Motherboard £34.99
1 x
Antec VP350P 350W 85% Efficient Continuous Power Supply £26.99
Total : £309.95 (includes shipping : Ex.VAT).
I have no idea why OcUK don't sell something like this as a home server setup