The Athlon 64 X2 5600+ chip can supprt A LOT of memory (RAM), but the limit it will also depend on the motherboard that you use.
According to Gigabyte's website for the GA-MA69VM-S2, it can support a maximum of 16GB across 4 slots. Which means, it can occupy up to 4 x 4GB DDR2 RAM.
Keep in mind, DDR2 being old technology and no longer mass producted as *much* it will be very expensive to buy brand-new. 8GB of DDR2 RAM will cost you EASILY 2x the price of 8GB of DDR3 RAM.
I never knew you can still buy socket AM2 / AM2+ motherobards. I mean, socket AM2 / AM2+ have been discontinued for well over 6 years.
If you are thinking of throwing quite a bit of money on this system (new motherboard, RAM, etc) then I strongly recommend that you build a new system altogether.
You can build a much more up-to-date a relatively powerful system starting at around $300 - $400 -- based on the socket FM2 / FM2+ APU's.
Phenom II X6 1090T @ 4.1GHz FX-8350 @ 4.8GHz // ASUS Crosshair V Formula 990FX // Sapphire Radeon HD Dual-X 7970 @ 1150/1500 // Thermaltake Frio w/ push-pull (using Antec Formula 7 Nano-Diamond + Cooler Master JetFlo fans) // 8GB (2 X 4GB) G.Skill RipJawsX 2133MHz // Corsair TX850 850W // Corsair Force GT 120GB SSD (OS) // Kingston V200+ 120GB SSD // WD Caviar Black 1TB // Windows 7 Ult x64