Eucalyptus is a popular tree grown for its timber as well as for its ornamental value. It has been grown in Southern California very successfully but is a native to Australia's rainforests. You can transplant your eucalyptus tree provided it has not grown too tall. One of the limiting characteristics of this tree is its long taproot. After a few years of growth, the taproot will be too deep to dig up.
Remove the tree from its present site. Don't expect to move a tree that is more than 4 feet high without having to do some major digging. The taproot and root system grows quickly and will be just as large as the tree is high. It is this characteristic that helps the eucalyptus tree survive harsh conditions and fires. Dig all around the tree and then down as far as the roots will grow.
Prepare the new site for your tree. Make sure you dig down as deep as the tap root is so it can continue growing straight down. Remove the soil and place it in a wheelbarrow. Add about half as much perlite to the soil and mix it together well. The perlite will hold moisture close to the roots, reducing some of the stress of the transplant.
Lift the tree out of its spot and lay it on its side on a tarp. Either drag it to its new spot, or wrap it up and transport it to its new site, sheltering it as much as possible.
Set the tree in the hole and hold it straight while you add the perlite and soil mixture. Tap it down as you go to make sure there are no air pockets that will dry out the roots. Add water after filling in half the hole. Let the water settle and continue filling in the hole.
Water the tree and shade it during the heat of the day. Transplanting is very stressful for the eucalyptus tree and it will need special care including shade and water until it is established. You shouldn't even attempt to move the tree in the middle of the summer.