But with the end of the old paper-based financial markets and the jobs they brought with them, such chances are rare. That is where colleges now come in.
The Securities and Investment Institute, a professional body for the financial sector, is launching a course to help school leavers, the unemployed and career-changers enter the City. Institute spokeswoman, Lora Benson said: "In the past people started as runners, but those jobs have disappeared with computerisation."
A pilot of the Introduction to Investment Award - part of an established City qualification - is running at four London colleges and has helped many of the 60 students who have done it find City jobs.
The institute plans to extend the course to 14 colleges across the country, to respond to the booming financial sector in cities such as Leeds and Manchester. Ms Benson said city firms had relocated many back-office jobs to the North, where there was now "great opportunity".
The short course, funded by the Learning and Skills Council, is running at Hackney Community college, Westminster Kingsway college, Tower Hamlets college and Kingston college.
The four colleges have also joined together to bid for a London Development Agency grant of nearly pound;400,000 aimed at preparing 200 for a City career.
Students learn about everything from shares to pensions to rules on insider dealing. The course will prepare them for a back-office role, supporting the traders or working with clients, or for a role as an independent financial adviser.
Ruth Martin, managing director of the institute, said: "We have to give employers a qualification they recognise as useful and relevant. It's a stepping stone."