Hi there,
Congratulations on the job first of all!
There are quite a few similar posts about working in Canary Wharf and looking where to live. Have a quick search for those!
In a nutshell, £150 per week isn't going to get very far if you choose to live relatively close by, however if you'll be doing a houseshare and will rent a bedroom with access to the the communal areas of a house, that may well be a suitable figure for many areas.
Have a look on Rightmove.co.uk and plug in your filter of £150 per week and check the returned results perhaps within a 2-3 mile radius around Canary Wharf. Evaluate that of course with your salary after tax to see how much of your disposable income you are comfortable with paying out for mandatory monthly living expenses. Overall, all of London, central and greater as well as the home counties will be commutable - I used to commute in when needed from Slough to Canary Wharf for instance and found I could do that within 75 minutes usually.
Travelling time, council Tax, bills, your social life travel and choices will all play a part and only you can answer what would suit you depending mainly on your realistic income levels.
Cheers, DtM! West London & Slough UK!