I found the costs stacked up very quickly when I got married, for sure. I definitely limited myself by wanting:
a) to get married in a pretty location with people at the ceremony,
b) to have a reception at the same location as the ceremony.
Because of UK marriage laws about where weddings can be held this really limited my options, and I went quite 'traditional' on some other stuff (i.e. clothes, flowers, photographer). In this sense, the answer to "is it possible to have a wedding cheaply?" is kind-of yes and no. It's 'yes' in that of course you can throw out almost all the rulebook and do it on a shoestring (nearly, see costs of ~£150 below). It's 'no' in that for specific aspects, if you choose to have them, it's really hard to get them below a baseline cost (e.g. if you want a professional photographer, there's only so low you can go on costs for that).
I reckon if you're willing to do a just-you-and-two-witnesses wedding followed by a party in a pub it could be done more cheaply, for sure. I think that in my area the bare minimum bureaucratic costs of getting legally married are ~£150. Many traditional/'proper' pubs around here (and I think Irish centres too) will allow use of function rooms for free on the assumption that people will be drinking and they will make their money that way, so if that fits your crowd, it might be a good plan. Almost certainly they won't allow outside catering so you might want to pick on the basis of their available food options.
That said, if you don't want to do a party, don't. No, you probably won't make yourself social pariahs. Certainly if you effectively 'elope', you sidestep the etiquette of 'who you have to invite' somewhat -- it's not offensive to people, in my experience, to be 'not invited' to a party that didn't happen the same way that it might be to be 'not invited' to a party that did happen. This is true of peers/friends, at least. Close family is pricklier but you will know best whether they will be upset at you not having a wedding.
Do you consider a wedding a need or a want?
For me, a legal wedding was a 'need' (for immigration reasons).
Once a legal wedding was in the works, a 'wedding' (e.g. having family and friends attend) was technically a 'want' but began to really feel like a 'need' because of how disappointing it would be to family otherwise.
Once having people in attendance was in the works, hosting them well was another 'want' that felt like a 'need', because a lot of them were travelling from another continent so I thought sandwiches at home didn't really make up for it.