Someone downloading the blockchain shouldn't make your computer unresponsive. If you don't have enough upstream bandwidth, it could make your internet really slow.
If you really want to run one, just use something like this:
https://vpsdime.com/cart.php?a=add&pid=416GB RAM
30GB SSD space
4 vCPU cores
2TB traffic
10Gbps uplink
1x IPv4
OpenVZ/Custom
$7/month
... wouldn't do the annual billing tho, since 30GB won't be enough for a year.
Another option is getting a decent refurbished desktop box with a reasonably modern processor and acceptable amount of RAM (~4GB should be fine). With that you can toss some linux or BSD on it and run bitcoind with the full transaction index and set yourself up a private electrum server.
The box will cost ~$150 to $200. For a bit more you can get a massive hard drive or an SSD to stick in it.
What this option gives you is a way to move the heavy lifting of handling the blockchain off of the box you interact with everyday while giving you a similar privacy benefit to running a full bitcoin node on it... As long as you set up electrum on you client machines to exclusively use your private bitcoin node/electrum server. You also won't have to worry about constantly worrying about overpaying for or exhausting your storage space as you would have to on a VPS.