If it is Eucalyptus gunni, yes you can - though I'd do it in the Spring. Actually, you an cut it to the ground and it will regrow - as a shrub.

If you have a wood burning stove it would keep you warm every year lol.

You can cut your Eucalyptus down now, taking 30ft. off will make it bleed, it should slow down before the spring. If you select buds and make a cut above, these buds will develop growth. Make a cut below the bud and they will not grow. Any you miss that do start growing can be rubbed out. This is a long term project and may not be succesful.