If you would like a modern computer in an embedded format (tiny board), try the PC Engines APU2/3. They offer an x86_64 quad-core, 4GB of RAM, SSD disk and three real network cards and consume around 12W-16W. In fact, this is a real computer. Works perfectly for Apache/database server. If you are buying a PC Engines board, buy directly from their shop, it is cheaper.
On the converse, to study system, ARM is very power efficient and works nice. I am running a couple of ARM boards (BBB, Raspberry Pie) and they would fit well in a firewall environment or to provide a service (DHCP, DNSSEC, etc ...). Though ARM boards lack a good SSD disc subsystem, that make them hard to use with databases and web servers. But this will come, hopefuly. The interest is the very-low consumption.
To play, you may also buy the older version of PC Engines board, the ALIX2D13, which can be found at low prices on eBay, but sells very quickly. These are very resistant boards, which cope with any climate, even hot-jungle-climate and are nearly unbreakable. They consume around 8W-12W. no SSD.
It is sometimes funny to study the price of electricity. In France, it is around 1W/month = 0,10 EUR. So 100W=10 EUR a month.
You may also look at Supermicro computers sold on eBay. Some sell at very low pricing and are monsters, with recent and decent hardware. Put an ARM board on their serial console and use them only on demand to lower consumption. Drop everything in a garage and you are done.
I agree that something can be done around ARM with plenty of RAM and Gig ethernet. But if you need a database server, just don't think about it. The APU2 board is around 100EUR and cost twice as much as a Beaglebone and is WAY more powerful for web server/database.
Therefore, depending on your needs:
* Internet/database => APU
* Network devices / Embedded projects => ARM
