Jesus told His disciples:

I and the Father are one. -- John 10:30

However, Jesus does not claim to be the Father. When Jesus says two separate persons are one, he means they are united or "joined together". A group of people could also be considered "one":

Holy Father, keep them in your name, which you have given me, that they may be one, even as we are one. -- John 17:11 (ESV)

The glory that you have given me I have given to them, that they may be one even as we are one, I in them and you in me, that they may become perfectly one, so that the world may know that you sent me and loved them even as you loved me. -- John 17:23

Jesus was not asking the Father to turn his disciples into gods. Just as the disciples are distinct individuals, yet united as one. So is Jesus and the Father distinct, yet united as one.

Humans can also be one with each other

One flesh

So the LORD God caused a deep sleep to fall upon the man, and while he slept took one of his ribs and closed up its place with flesh. And the rib that the LORD God had taken from the man he made into a woman and brought her to the man.

Then the man said, “This at last is bone of my bones and flesh of my flesh; she shall be called Woman, because she was taken out of Man.”

Therefore a man shall leave his father and his mother and hold fast to his wife, and they shall become one flesh. And the man and his wife were both naked and were not ashamed.

-- Genesis 2:21-24 (ESV)

Also, Jesus quoted this:

But Jesus responded, “He wrote this commandment only as a concession to your hard hearts. But ‘God made them male and female’ from the beginning of creation. ‘This explains why a man leaves his father and mother and is joined to his wife, and the two are united into one.’ Since they are no longer two but one, let no one split apart what God has joined together.” -- Mark 10:5-9 (NLT)

Or do you not know that he who is joined to a prostitute becomes one body with her? For, as it is written, “The two will become one flesh.” -- 1 Corinthians 6:15 (ESV)

One spirit

But he who is joined to the Lord (Jesus Christ) becomes one spirit with him. -- 1 Corinthians 6:17 (ESV)

One soul

If your brother, the son of your mother, or your son or your daughter or the wife you embrace or your friend who is as your own soul... -- Deuteronomy 13:9 (ESV)

As soon as he had finished speaking to Saul, the soul of Jonathan was knit to the soul of David, and Jonathan loved him as his own soul. And Saul took him that day and would not let him return to his father’s house. Then Jonathan made a covenant with David, because dhe loved him as his own soul. -- 1 Samuel 18:1-3 (ESV)

Now the full number of those who believed were of one heart and soul, and no one said that any of the things that belonged to him was his own, but they had everything in common. -- Acts 4:32 (ESV)