Can the bailiff take my car? It's not a work car, but without it I would not be able to get to work?

Yes

Exempt goods include vehicles which are necessary for use personally by the debtor in his employment, business, trade, profession, study or education, up the value of £1,350.^[1]

The regulations do not provide for making vehicles used for travelling to and from work to be exempt. It is only exempt if the vehicle is used in the debtors employment.

If the vehicle is used in the debtors employment, and its value is under £1,350, the debtor may make a claim to exempt goods.^[2][3]

^{[1] Regulation 4 of the Taking Control of Goods Regulations 2013

[2] Civil Procedure Rule 85.8

[3] See Bailiffs took your exempt car or goods}