• The hospital is near.

• The hospital is close.

Both these sentences are correct and mean the same thing, a SHORT DISTANCE.

NEAR and CLOSE have also this same meaning when referring to TIME:

• Summer is near.

• Summer is close.

When close has this meaning, it is pronounced /kləʊs/.

And notice that close is often followed by the preposition to:

• The hospital is close to the park.

• The hospital is near the park.

Be Careful! Don’t confuse the adjective close with the verb close /kləʊz/. If you close something, you move it so that it fills a hole or gap.

Close near

• My friend and I live in different countries, but we are very close.

Here, I am saying that my friend and I are not in the same physical area, but we are emotionally connected. We are good friends.

However, if I say, “My neighbor and I live in the same apartment building, but we are not close,” I mean we occupy the same physical area, but we are not emotionally connected. My dog tried to bite his dog once, and he has never spoken to me again!

You can refer to someone you know well as a close friend.

• Mike and I are very close friends.

• His father was a close friend of my father.

You can refer to someone who is directly related to you as a `close relative.

• She had no very close relatives.

You can also refer to someone as a `near relative, but this is less common. 

You can say:

• They’re a close family.
• My dad was closer to his brother than to his sister.

Close is also used in the following collocations:

• a close encounter
  The mountaineer Joe Simpson has had several close encounters with death.
• a close race
  After a close race, Obama won the election.
• a close finish
  It was a close finish – only a tenth of a second separated the two runners.

Near close

• a near miss
  The asteroid passed 27,700 km from the surface of Earth – a near miss.
• in the near future
  The volcano could erupt in the near future, according to scientists.
• in the near distance
  We could see someone in the near distance.

https://grammar.collinsdictionary.com/english-usage/what-is-the-difference-between-near-and-close

Near and nearby both mean close. They are just used differently in the sentence.

Here are some guidelines:

It is what a PREPOSITION does. A preposition is a word that shows the relation of one word in a sentence to another word.

• There is a beach near my house.  (this means that there is a beach close to my house)
• He worked in a restaurant near the station. Not: He worked in a restaurant nearby the station.

NEARBY can never be properly used as a preposition, although you may hear someone say it occasionally:

• The book is nearby the table. (it is not considered correct)
• The book is near the table. (near means close to the table)

• We can meet at a nearby beach.
• Luckily, the nearby buildings weren’t damaged by the fire.

• The beach is nearby.
• Does Paul live nearby? 

Remember that an ADVERB is a word that tells us more about a verb, an adjective or another adverb. It describes or modifies these words.

So, note that NEARBY can appear either before or after a noun that it describes, but NEAR can normally appear only before the noun, and even then, only when referring to time:

• We slept at a nearby motel. (nearby + noun)
• We slept at a motel nearby. (noun + nearby)
• I hope to visit you in the near future. (near + noun, and near=close in time)

So, NEAR can mean close in time (=soon), as well as close in distance, but nearby cannot, as shown below:

• Summer is near. [near=close in time]
• May is nearby.  [nearby cannot mean close in time]

According to Practical English Usage by Michael Swan, near to is also possible with the same meaning as near, but it is less common.

• There is a beach near (to) my house.
• I came near (to) hitting him. (prep + verb + ing)