From a Biblical perspective, the only way a person can be saved is if they believe in the Son of God. So that means that there is no salvation for those who do not believe. This is what the Bible states:
For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life.
For God sent the Son into the world, not to condemn the world, but that the world might be saved through him.
He who believes in him is not condemned; he who does not believe is condemned already, because he has not believed in the name of the only Son of God.
And this is the judgment, that the light has come into the world, and men loved darkness rather than light, because their deeds were evil.
For every one who does evil hates the light, and does not come to the light, lest his deeds should be exposed.
But he who does what is true comes to the light, that it may be clearly seen that his deeds have been wrought in God. (John 3:16–21)
The reality is that it is impossible for a person not to do evil at some point. Even to doubt the integrity of God is evil. Therefore to believe that the Creator does not exist is evil.
What is evil? Evil occurs when we deny the natural rights of another.
To put this in a human context, evil occurs when a child disobeys its parents or when a parent neglects to adequately provide for a child.
Some people like to think that they have never been evil. Personally, I have yet to find such a person who is morally perfect and who has never committed any evil, even though I have known some people who are close to perfect. But being nearly perfect is not the same as being nearly evil.
The suggestion that a person might be nearly evil sounds ludicrous, for a person is either evil or morally perfect. But being nearly evil simply means having the capacity to do evil, only the person has not done evil. Once evil has been committed, the person who commits evil can never be morally perfect by anything the person does.
To use an analogy: for anybody to be nearly perfect is the same as not possessing the ability to be light and relying upon the light of other sources to see in the dark. If one is the light, darkness is dispelled without the need of another source to see where one is going.
From what the Bible teaches, we could say that the first man and woman created were nearly evil because they had yet to commit evil. At the same time, they were nearly perfect because, even though they had never sinned, they still did not possess the ability to live forever.
The equation: an evil thought plus action multiplied by sin equals death still stands. Or as the Apostle James puts it:
Each person is tempted when he is lured and enticed by his own desire. Then desire when it has conceived gives birth to sin; and sin when it is full-grown brings forth death. (James 1:14–15)
The question you ask is Can one be saved if he only accumulates good deeds without doing evil but does not believe in God?
The above is predicated upon the assumption that you mean saved from death, which means eternal banishment from the presence of God and the thrill of being a dill on one’s own for eternity.
Lonesomeness has it problems. Inner loneliness is terrifying. Inner loneliness for eternity is not something one wants to contemplate, but this is the only option for those who reject the bliss of God’s presence. I have experienced the terror of inner loneliness for a brief moment. If not experiencing the terror of inner loneliness for eternity is what you mean by being saved, then you need to know God and have a relationship with Him.
However, you did not say from what you were expecting to be saved if you did not believe in God.