Go for it, hopefully my story will give you the inspiration you need, but be warned this stuff becomes addictive!
I am 6'4" and back in January 2003 aged 33 I weighed in at a mighty 18 stone. With some work colleages we had a challenge to lose 10% of our body weight by easter & in order to achieve this I decided to try running.
When I started I couldn't run 1km without stopping for a rest, but in a few months the distances gradually increased so I could run 10k (the distance from home to work) in less than an hour, so I started to run to work a couple of mornings a week. By Easter I was down to 15 stone & that is where I have stayed (+/- 5lbs).
Then a friend bet me that I couldn't run a marathon in less than 4.5 hours. So I started training & ran the 2004 London marathon in 4hrs 9 minutes. At the time this was the most physically painful but most rewarding thing I had ever done. Bitten by the bug I ran the marathon in 2005 & this year but have not improved on my time (injuries & illness were factors both times).
Driven by a mid life crisis of wanting to do an Ironman before I am 40 I entered the Olympic Distance at the London Triathlon this year to see how I would get on.
My training programme was probably a little short as I concentrated on the marathon until April & then took a whole month off to recover. When I started I found I could only swim 100m before I needed a rest & I was taking over 1.75 hours to cycle 40k. I made gradual progress & by the end of July I could happily cycle 40km on a Sunday afternoon in 1.5 hours & swim 1500m in 35 minutes in a pool once or twice a week.
I arrived at Excel on Sunday absolutely terrified, I had never swum in open water & never covered the race distance in training. My stated aim was to finish, secretly inside I wanted to be under 3 hours, but I knew there was a chance I might not even make it to the finish line.
As it turned out I did better than I could ever have hoped for. Swim 31:58, Bike 1:20:55 & Run 57:26 plus transitions for a total time of 2:57:26. I too was forced to walk a couple of times on the run but it was very hot & I knew I need to drink.
As I reflect on what I have achieved, I was 365th out of 504 finishers in the 35-39 age group so 72% of the people were faster than me BUT I am 6'4" & 210lbs which is not really the ideal build for an athlete, 3 years ago I struggled to run for a bus & this was my first attempt at triathlon.
As for damaging your bank balance, I tried to keep costs down by buying a bike off ebay for £250 & renting a wetsuit. On the day I passed people on far more expensive bikes than mine & was passed by people on older cheaper ones, maybe spending thousands on equipment helps if you are at the sharp end of the race but it cannot be a substitue for training.
So in summary, I am 1" taller than you & exactly the same weight, my 10k pb is 51mins and I managed to finish the olympic distance in under 3 hours so I am sure you can too.