Update: Since the original post it has been confirmed that the cause is a DDoS attack. WyomingMyst stated:
Okay, I have received confirmation on my end, yes this is a DDoS attack.
The best course of option is to avoid playing Blizzard games until the attack is mitigated. Again, Competitive Penalties cannot be reversed. I am going to go AFK (because I am really hungry and I was about to go get food right now this started to happen), but I will try to post updates as I get them.
Original: Confirmed by a post on the official Blizzard forums, Overwatch and other games accessible via Battle.net are currently experiencing server issues resulting in several players being discontinued during matches.
WyomingMyst has even highly recommend that players avoid competitive play in part due to the fact that “Penalties cannot be reversed from disconnections even if they are due to server issues.”
Shortly after the forums post was made, WyomingMist was able to confirm that any site with the battle.net domain (including support and the shop) is also affected.