Starting position is certainly a factor, but I think it only becomes a significant, ongoing disadvantage if you don't attempt to use some of the in-game systems that are designed to help you catch up. Also, if you let an enemy civ become an end-game 'monster', then whose fault is it, really? Often, AI's will notice a growing threat, and even ask for help in going to war (with said threat); but if you stick your head in the sand, and ignore the appeals of other civs (or if you yourself are just not paying attention to global trends), you only have yourself to blame later ... after Hiawatha, Washington, etc. has taken over the entire map. A bad starting position may be challenging, but it's not necessarily an instant-loss. In fact, some of my best experiences in Civ V are with games that had very bad starting positions.
There are numerous ways to offset a bad starting position, some ways based on supplementing civ advancement, others based on supplementing the war game (which is itself, a form of 'equalizing the competition'). Alliances with city-states provide a system for circumventing resource deprivation; espionage provides a system that allow you to steal techs and rig city-state elections (which, in turn, swing needed resources back to you); religion provides a system for supplementing your civ in areas where it may be lacking (that's why there are so many beliefs to choose from); prize ships allow you to build a fleet from virtually nothing (same goes with certain UA's, like Furor Teutonicus, Barbary Corsairs, etc.); the diplomacy and economic systems allow you to manipulate other civs into warring with each other; the diplomacy and trade systems provide hideous ways to fleece the civs of their gold; etc.
I would suggest trying to play something like a 3-city culture game, where you're constantly on the defensive and are forced to "play from behind". You'll learn a lot about diplomacy, economic manipulation, and waging a defensive war.
Also, watch Maddjinn's Deity Dido LP series; he starts the game in a horrible starting position, but utilizes the game's many built-in systems to work his way to a victory. It's certainly one of the better LP's out there that showcase multiple ways of manipulating bad starting circumstances.