The simple answer is "yes". The complicated answer is "as long as the server is set up correctly". Basically, install the Ubuntu Server version you want and set it up. Without knowing your specific home network configuration, it is hard to give you specific details of how to sign in to the server. However, I can attempt to give you same basic details.
1) Install the server
Fairly self-explanatory here. Install it, create the user accounts you need (I suggest having separate user accounts for the home user and the remote user).
2) Set up your laptop.
Install whatever flavor of Ubuntu you want on the laptop. The most current LTS is 14.04. If you have older hardware, you may consider using 12.04 LTS.
3) Set up your network
I am going to make some assumptions here. You have a modem that is hooked up to a router (or a combined modem/router) for your home network. You will need to forward the ports for SSH. The standard is 22, but I suggest you change that. Open a terminal and do this:
su #you will need to enter
#your root password after this command
nano /etc/ssh/sshd_config #you could also use vi
You should see something similar to this:
# Package generated configuration file
# See the sshd_config(5) manpage for details
# What ports, IPs and protocols we listen for
Port 22 #<- change this line to some other port (such as 6022)
# Use these options to restrict which interfaces/protocols sshd will bind to
Then restart the service:
service ssh restart
Remember that port. You will need it.
4) Let's attempt to log in.
On your laptop, make sure you are connected to the internet and issue this command from a terminal:
ssh -p [port_number_you_entered_in_step_3] [ip_address_of_server] -l [username]
Without the brackets, of course. If you set it up correctly, you should be able to log in to the server. Once you are outside of your network, you can do the same thing, but you will need to change the ip address to your public IP address (which you can find by typing "what is my ip address" into a Google search from your home compuyter).
That should cover the basics. Past that, you should ask any further questions over @ https://serverfault.com/.