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I have the following Java-related packages installed on my Debian 8 machine

# sudo dpkg -l | grep -i java
ii ca-certificates-java 20140324 all Common CA certificates (JKS keystore)
ii java-common 0.52 all Base of all Java packages
ii libatk-wrapper-java 0.30.5-1 all ATK implementation for Java using JNI
ii libatk-wrapper-java-jni:amd64 0.30.5-1 amd64 ATK implementation for Java using JNI (JNI bindings)
ii openjdk-7-jre:amd64 7u181-2.6.14-1~deb8u1 amd64 OpenJDK Java runtime, using Hotspot JIT
ii openjdk-7-jre-headless:amd64 7u181-2.6.14-1~deb8u1 amd64 OpenJDK Java runtime, using Hotspot JIT (headless)
ii tzdata-java 2018e-0+deb8u1 all time zone and daylight-saving time data for use by java runtimes

I just want to install Java 8 but I keep running into dependencies issues:

# sudo apt install openjdk-8-jre
Reading package lists... Done
Building dependency tree 
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 openjdk-8-jre : Depends: openjdk-8-jre-headless (= 8u171-b11-1~bpo8+1) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.


# sudo apt install openjdk-8-jre openjdk-8-jre-headless
Reading package lists... Done
Building dependency tree 
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 openjdk-8-jre-headless : Breaks: ca-certificates-java (< 20160321~) but 20140324 is to be installed
E: Unable to correct problems, you have held broken packages.

How can I upgrade to Java 8, addressing all dependencies issues at the same time?

You need to tell apt to consider backported packages:

apt -t jessie-backports install openjdk-8-jre