To get YaCy running, the only thing you need is a somewhat recent computer (everything that used to be up-to-date in this millennium should be all right) and an operating system that supports Java. Apart from this, you should allow for 25GB (or at least 1-2 GB for a start) of disk space on your hard-drive for collected data on websites and the index itself (keep in mind in which directory YaCy stocks its data, e.g. in /var/lib for Debian GNU/Linux). The more physical memory (RAM) you have the better, but 256MB are a good start. Less is also acceptable, but it will make work a bit slow.
YaCy has been written in the Java programming language, which means that it is available for a great number of computer systems, as Java is available for almost all systems in use. An outstanding feature of Java programs is the fact that they can be run on different systems without any modification.
Java can be downloaded and used for free. If there is no Java environment on your system yet, you must install it before installing YaCy. GNU/Linux distributions may include, e.g. a free one called OpenJDK. Otherwise, Java is available from the Sun website. The minimum Java version you need for YaCy is Java 7 — which might change in the future.
Note that Apache Solr beeing a YaCy core component, it is a good idea to follow also Solr recommendations. For example, YaCy 1.82 includes Solr 4.10.3 : Java 1.7.0_u55 or later is recommanded (see Solr System Requirements)
Because of this — and the fact that the newer Java version is more powerful than the old one — you should chose Java 7 right from the start. If the only thing you want to do is just run Java programs you can make do with the JRE (Java Runtime Environment). If you want to develop Java programs or even help making YaCy better, you will need the JDK Java Development Kit.