I kind of got into it by accident ... I'm a trained public sector accountant but hated the sheer soulless grind of it and switched to customer service roles after my youngest was born because I wanted more contact with people. I'd done that for years but then got stuck in a job for a company I hated after I had my cancer and no-one wanted to take a chance on a new employee who might end up on long term sick
Then, when my husband returned to work (he stayed at home and raised the children because childcare for 3 was prohibitive) he started at a tiny, independent, very local FD as a driver/bearer and, as it happened their main office administrator/funeral director was looking to retire but they had had no luck finding someone who could do the job. Because it's such a small company, the role basically includes a bit of everything - from hair and make-up for the deceased to dealing with bereaved families, to the (hand written, double entry!) accounts, wages etc and all stops in between. Finding someone with all the skill sets was apparently a bit of a challenge - people were either comfortable with the physical and emotional side but not up to the accounts and admin (there is a lot of paperwork associated with funerals) or vice versa.
Now I'm not squeamish - my Mother was a geriatric specialising SRN and I'd been raised with death as an un-remarked fact of life (plus grew up in a farming community) and had been dealing with customers in all kinds of stressful situations for years and obviously the admin and accounts weren't a problem, so I was lucky enough to just walk into the job. A 'right place/right time' stroke of serendipity
The bigger companies do often offer training places though - especially the Co-op and Dignity groups - and they are less likely to involve actual hands on work with the dead as their staff tend to specialise in particular areas like Funeral Arranging. You're good with people, empathic, patient and have a great sense of humour (pretty much essential that last one! Surprisingly many families do appreciate a little levity at appropriate points) so I think it's something you could definitely do
The NAFD also do part time courses that are open to those not currently working in the profession too, so they might be worth a look if you're interested
Overall I love my job ... yes, working for a very small company where I'm pretty much the main resource can be a little exhausting in terms of time off etc, but that's more to do with the size of the outfit rather than the work itself and wouldn't be any different if we were engaged in a different field. The work itself however is emotionally very rewarding - I never have a day when I go home thinking 'well that was a waste of 8 hours of my time', and I love knowing that I've done something, no matter how small, to do those last things for people's loved ones and to ease people through hard emotional times.