Two recent bad starts:
Started on a desert tile (yes, my starting position was a DESERT tile). A couple of plain tiles below that. Everything else desert. No fresh water in sight. Moved my worker a bit SW. Encountered a large swathe of tundra. Gave up on the game at this point. This is completely unmodded vanilla Civ.
The my next worse start - I actually played this one. It turns out my continent was entirely desert, tundra or mountains except for two narrow strips of green separated by a huge desert. Here's the continent:
(1/3) Tundra
Strip of Green
Large desert
Strip of Green
Large desert
All mountains
One grassland tile at the very tip.
To make things worse there are NO horses OR iron ANYWHERE on the continent.
There was ONE source of freshwater on the entire continent and it wasn't on my strip of green (the other strip was bigger too). Once I worked this out I prepared for a massive rush of the English who owned the freshwater source. I was also hunting for iron/horses at this point as I didn't realise there weren't ANY on the continent. While building archers, I got my workers to build a road across the large desert seperating us. After over-running the English, I used the workers to do a massive irrigation chain right across the desert to my core cities.
After working out there was NO iron/horses on the entire continent I rushed towards map-making as quickly as possible (as you can see I was pretty behind in tech), built a galley. Luckily the grasslands tile at the tip was separated from another continent by one water tile. So I used the galley as a land-bridge. EDIT: Forgot to mention, at this point the much more advanced Indians landed a settler on my continent and founded a city right in the path all my settlers and troops were headed to to get to the new continent. Their cultural borders expanded so to form a chokehole. Everytime I tried to move my troops across so I could get to my landbridge they kicked me out. So I had to fight another early war (this time with India). Finally found iron (and horses and something besides tundra/desert!). I don't have the tech to connect harbours though so this is useless for my core cities. The sight of all that green was heavenly. Until I discovered all the barbarians living there. My first settler gets killed. As soon as I found my first city, "A Massive Barbarian Uprising" occurs next to it. As I have one puny regular archer guarding it, I move the archer out of the city. Sure enough, all of the huge stack of horsemen go for the (size 1 newly founded) city and carry away some gold. The barbarian problem is so bad that I have to send every worker out with a guard otherwise they don't last long. However, there is NO freshwater on the entire half of the new continent. I finally find some freshwater halfway through the continent, but it is controlled by the Chinese (who it turns out had a wonderful start and are much more advanced than me). I guess you could say this is the worse start that I actually played.
