The AS application is self-contained, but you do need a JVM. I think a JRE is sufficient, but we've used the full OpenJDK for deploying. The AS docs specify version 1.8 for your java.
cd /vhosts/ && wget https://github.com/archivesspace/archivesspace/releases/download/v3.0.1/archivesspace-v3.0.1.zip
unzip archivesspace-v3.0.1.zip
For anything other than local fun, use a robust SQL ("squill") database. See the official readme for different info. Note that the following steps assume that you have already downloaded and unpacked the ArchivesSpace release from github.