One forgotten part of the reason why the Jews were called "the chosen people" was that they were called to go and tell the people of the world about God and His redemption plan.
The Temple, with it's Holy of Holies was built as a dwelling place for God amongst men. It's design and it's various areas and it's various purpose built furniture and implements, were there to reveal God's character to mankind.
Now, there were actual a series of Temples in the Old Testament, as each one was destroyed, another was built to replace it.. the one in question at the time of Christ and therefore of the visit of the Eithipian Eunoch, was Herod's Temple, which was destroyed by the invading armies of Rome, in AD 70.
There was in each Temple an important area known as the "Court of the Gentiles". It was not considered holy ground, and non-jews were allowed there. Here buying and selling went on, and it was here that Jesus clensed the temple as described in John 2:14-17.
Certainly, Gentiles were prohibited from going into the inner courts of the Temple, but there was certainly a large group of God fearers amongst the nations and they were attracted by the Jewish religion and it's strong moral code. They attended synogogue services and worshipped the Jewish God, but did not follow all the Jewish Laws, for example those relating to circumcision and diet.
God, through Isaiah said ""And foreigners who bind themselves to the LORD
to serve him, to love the name of the LORD
and to worship him,
all who keep the Sabbath without desecrating it
and who hold fast to my covenant--
these I will bring to my holy mountain
and give them joy in my house of prayer.
Their burnt offerings and sacrifices
will be accepted on my altar;
for my house will be called
a house of prayer for all nations."
Isaiah 56. 6-7
this is where the Eithiopian was going.. .. he was going to worship as a God Fearer. This is the reason why on the day of Pentecost, as the Holy Spirit descended on the believers, everyone from around the world, who were gathered in Jerusalem, "Parthians, medes, Elamites, residents of Mesopotamia, Judea and Cappodocia, Pontus and Asia, Phyrigia, Pamphylia, Egypt and the parts of Libya near Cyrene, visiors from Rome, Cretans and Arabs.. God fearing Jews from every nation heaven". (Acts 2) heard the Gospel preached in their own language.