Hi Mikele and welcome to GoY. Very bad idea, especially as a Eucalyptus can grow up to 12 feet in a year! Unless you have a very large garden a euc. is rapidly going to outgrow its space. Only alternative would be to grow in a pot but if it is one of the big eucs. that won't work - sorry.

It looks right next to the house, not 6ft away in the photo. They suck moisture out of the soil and they grow more rapidly in the UK (prob same in France) due to the higher rainfall than their native home. That leaves them vulnerable in the wind as they tend to grow rapidly and thin and have a name in the UK for falling down becuase of that top heaviness.

I love their scent and the juvenile leaves. Cutting it back every few years would give you both but still needs to be more than 6ft away, I would assume the roots would be less of an issue if regularly pruned?

If this is the florist's type, you might get away with it by pruning hard several times a summer, to keep it small and in juvenile foliage. Just remember that if maintenance isn't done religiously, it will demolish your house!! :/

It's a Thug and will do damage to your property, I had one at the bottom of my garden and it was breaking up my neighbours brick built workshop, My Brother took it down for me but the stump started to put up shoots, I kept pulling them off till it died. Get Rid Of It.
Garden Centres don't Warn you about these trees do they.

Thank you all for your comments,I think I will very soon remove it.

Definitely sooner rather than later!

Before we had a phoenix palm in this location, it was ravaged by the moths that have been killing them across Europe...anyway we wanted to replace it with a tree that stays green all year but does not shed needles, as we already have a mediterranean pine that does just that. Is there an evergreen you would recommend that could grow next to the house without damaging it?

Some of the smaller columnar Junipers could do it, such as 'Robusta Green' or 'Blue Point'. 'Wil Fleming' Yaupon (Ilex vomitoria) could also work, if your winters aren't too cold.

If you're looking for a nice evergreen shrub "Sophora Sun King" is lovely :O))