There are three main options for staying as close as possible to the Vatican city:
Stay at the only hotel in the Vatican
If you want to be close to the Vatican, you will not find somewhere much closer than Residenza Paolo VI. The hotel itself was built on the site of an Augustinian Monastery, and provides a small sanctuary away from the chaos of Rome.
This is the only hotel in the Vatican, and it has the easiest access for visiting the Vatican museums, Castel St Angelo and St Peter's Basilica, as well as all the other major attractions in Rome if you follow the main roads down away from the Vatican.
The hotel includes a bar, restaurant, and buffet breakfast, and if you eat on the third floor restaurant you will have stunning and exclusive views overlooking St Peter's square. The hotel is advertised as a boutique hotel with only 35 rooms, and there is an emphasis on style and comfort. The hotel is relatively new, only inaugurated in the year 2000, and has all the usual comforts of a good hotel while maintaining warm, atmospheric furnishings.
Stay in a Vatican monastery
Staying in a monastery can be a unique and relatively inexpensive accommodation option in Rome, even if you are not a Catholic. There are hundreds of religious "houses of hospitality" in Rome, but most will include meals in the price, and safe, clean, and comfortable rooms.
One thing that might put people off staying here is that there is a nightly curfew in some places, but if this is not an issue then staying in a monastery is a wonderful alternative to staying in a hotel for your visit to the Vatican.
Casa Bonus Pastor (No nightly curfew)
This place is owned by the Vatican, and it is a short walk up to the Vatican city if you choose to stay here. Casa Bonus Pastor has clean comfortable rooms (with single beds only), has no nightly curfew, and meals can be included in the price.
All of the rooms are clean, comfortable and quiet, and the food is authentically Italian and has generous servings. What makes this place special is that it is surrounded by a luscious green park, and it has plenty of parking.
Casa Santo Spirito (nightly curfew)
This house is owned and run by the Franciscan Sisters of the Sorrowful Mother, and it offers more of a hostel experience. They practice religious hospitality, and this is a wonderful place to use as a base for your pilgrimage in Rome.
The house offers shared, dorm-like rooms (there is one private room) that each have a bathroom. Breakfast is included in the price, and they accommodate all kinds of travellers. The hotel itself is overlooking St Peter's Square, and it has several common areas to hang out in while you are in the convent itself.
Stay in Prati neighbourhood close to the Vatican
If you are wanting to book accommodation that is close to the Vatican, you will want to explore accommodation options that are in the Prati district. This is the neighbourhood that is just north of the Vatican, and it is well connected to the rest of the city via Metro.
Prati is not a very popular tourist area, so it is a little off the beaten-track and you could therefore escape the crowds a little. The area itself is characterized by wide, tree-lined boulevards and quite a lot of shopping areas and good restaurants. There are several hotels in this area, but it would also be a great place to stay if you wanted to rent a private apartment.
As well as being close to the Vatican, you have access to Rome's largest market, the Mercato Trionfale. If you are a foodie traveller, then having space to cook with ingredients from the local market would be a dream.