Run the following commands as a user with sudo privileges or root to update the packages index and install the OpenJDK 11 JDK package:

Copy$ sudo apt update
$ sudo apt install openjdk-11-jdk

Once the installation is complete, you can verify it by checking the Java version:

Copy$ java -version

The output should look something like this:

Copyopenjdk version "11.0.7" 2020-04-14
OpenJDK Runtime Environment (build 11.0.7+10-post-Ubuntu-3ubuntu1)
OpenJDK 64-Bit Server VM (build 11.0.7+10-post-Ubuntu-3ubuntu1, mixed mode, sharing)

Set JAVA_HOME Environment Variable: OpenJDK 11 is located at /usr/lib/jvm/java-11-openjdk-amd64/bin/java

Once you found the path of your preferred Java installation, open the /etc/environment file:

Copy$ sudo nano /etc/environment

Assuming you want to set JAVA_HOME to point to OpenJDK 11, add the following line, at the end of the file:

CopyJAVA_HOME="/usr/lib/jvm/java-11-openjdk-amd64"

For changes to take effect on your current shell you can either log out and log in or run the following source command:

Copy$ source /etc/environment

Verify that the JAVA_HOME environment variable was correctly set:

Copy$ echo $JAVA_HOME

You should see the path to the Java installation:

Copy/usr/lib/jvm/java-11-openjdk-amd64

for reference you can follow this link below How to Install Java on Ubuntu 20.04

I wanted to clarify that as of 9 December 2016, you most certainly can install Java 8 on Ubuntu Bash for Windows 10 and that @Karl Horton is correct.

You will need to install unzip sudo apt-get install unzip

Copy this script somewhere in your bash for windows session and make it executable (chmod +x filename). If you do not use a command line based editor such as vim then you will have windows line endings to deal with. you can use dos2unix or your preferred way of dealing with that. I just paste it into a file using vim.

 #!/bin/bash

set -ex

# UPDATE THESE URLs
export JDK_URL=http://download.oracle.com/otn-pub/java/jdk/8u131-b11/d54c1d3a095b4ff2b6607d096fa80163/jdk-8u131-linux-x64.tar.gz
export UNLIMITED_STRENGTH_URL=http://download.oracle.com/otn-pub/java/jce/8/jce_policy-8.zip

# Download Oracle Java 8 accepting the license
wget --no-cookies --header "Cookie: oraclelicense=accept-securebackup-cookie" \
${JDK_URL}
# Extract the archive
tar -xzvf jdk-*.tar.gz
# clean up the tar
rm -fr jdk-*.tar.gz
# mk the jvm dir
sudo mkdir -p /usr/lib/jvm
# move the server jre
sudo mv jdk1.8* /usr/lib/jvm/oracle_jdk8

# install unlimited strength policy
wget --no-cookies --header "Cookie: oraclelicense=accept-securebackup-cookie" \
${UNLIMITED_STRENGTH_URL}
unzip jce_policy-8.zip
mv UnlimitedJCEPolicyJDK8/local_policy.jar /usr/lib/jvm/oracle_jdk8/jre/lib/security/
mv UnlimitedJCEPolicyJDK8/US_export_policy.jar /usr/lib/jvm/oracle_jdk8/jre/lib/security/

sudo update-alternatives --install /usr/bin/java java /usr/lib/jvm/oracle_jdk8/jre/bin/java 2000
sudo update-alternatives --install /usr/bin/javac javac /usr/lib/jvm/oracle_jdk8/bin/javac 2000

sudo echo "export J2SDKDIR=/usr/lib/jvm/oracle_jdk8
export J2REDIR=/usr/lib/jvm/oracle_jdk8/jre
export PATH=$PATH:/usr/lib/jvm/oracle_jdk8/bin:/usr/lib/jvm/oracle_jdk8/db/bin:/usr/lib/jvm/oracle_jdk8/jre/bin
export JAVA_HOME=/usr/lib/jvm/oracle_jdk8
export DERBY_HOME=/usr/lib/jvm/oracle_jdk8/db" | sudo tee -a /etc/profile.d/oraclejdk.sh

And now I can do the following

fieldju@DESKTOP-LTL6MIC:~$ java -version
java version "1.8.0_112"
Java(TM) SE Runtime Environment (build 1.8.0_112-b15)
Java HotSpot(TM) 64-Bit Server VM (build 25.112-b15, mixed mode)                                                

The links and versions in the above script are likely to be out of date by the time you read this, so just head over to http://www.oracle.com/technetwork/java/javase/downloads/index.html accept the license so that their js lets you copy the new URLs and you should be good to go.

Here's how I just made Java "just work" for my WSL installation.

ln -s /mnt/c/Program\ Files\ \(x86\)/Java/jre1.8.0_151/bin/java.exe /bin/java

What this does is create a symbolic link in your /bin folder to your Windows Java binary. Whenever WSL tries to invoke Java now, it's redirected to the Windows version. You will probably need to adjust the path as Java versions change in the future.