ticket for vatican and sisteine chapel

You can do so online at this site, the Vatican's Ticket Office.

As previously mentioned, you'll need to book your reservation at the Vatican Museum's website. Please see details from Rick's Rome guidebook below:
Reservations: The Vatican Museums can be extremely crowded, with waits of up to two hours to buy tickets (figure about a 10-minute wait for every 100 yards in line). Bypass the long ticket lines by reserving an entry time at http://mv.vatican.va for €20 (€16 ticket plus €4 booking fee). It’s easy. Just choose your day and time, then check your email for your confirmation and print out the voucher. At the Vatican Museums, bypass the ticket-buying line and queue up at the “Visitor Entrance with Online Reservations” line (to the right). Show your voucher to the guard and go in. Once inside the museum, present your voucher at a ticket window (cassa), either in the lobby or upstairs, and they’ll issue your ticket. Even with a reservation, read “When to Go,” next.
When to Go: Except in winter, the museum is generally hot and crowded, with long ticket lines and shoulder-to-shoulder sightseeing through much of it. The worst days are Saturdays, the last Sunday of the month (when it’s free), Mondays, rainy days, and any day before or after a holiday closure. Mornings are most crowded. The best (or least-worst) time to visit is a weekday after 14:00--the later the better. Another good time is during the papal audience on Wednesday morning, when many tourists are at St. Peter’s Square (the only drawback is that St. Peter's Basilica is closed until roughly 13:00, as is the exit to it from the Sistine Chapel).
Regarding visiting the Sistine Chapel, reservations are not required or taken--please see the following from Rick's Rome guidebook:
Hours: The church is open daily April-Sept 7:00-19:00, Oct-March 7:00- 18:30. It closes on Wednesday mornings during papal audiences (until roughly 13:00).
The dome (cupola) is open to climbers daily from 8:00; if you're climbing the stairs all the way up, the last entry time is 17:00 (16:00 Oct-March); if you're riding the elevator, you can enter until 18:00 (17:00 Oct-March).
The Treasury Museum is open daily April-Sept 8:00-18:50, Oct-March 8:00-17:50.
The crypt (grotte) is open daily 9:00-16:00.
Avoiding Lines: There’s no surefire way to avoid the long security lines. The checkpoint is typically on the north side of the square, but is sometimes closer to the church or tucked under the south colonnade.
St. Peter’s is often accessible directly from the Sistine Chapel inside the Vatican Museums--a great time-saving trick, but unfortunately not a reliable one.
Avoid the worst crowds by visiting before 10:00. If you come after 16:00, crowds are lighter, sunbeams can work their magic, and the 17:00 Mass (Mon-Sat) in the apse fills the place with spiritual music. But after about 16:00, the crypt is closed, and the area around the altar is often roped off to prepare for Mass.
Dress Code: No shorts, above-the-knee skirts, or bare shoulders (this applies to men, women, and children). Attendants enforce this dress code, even in hot weather. Carry a cover-up, if necessary.

I don't know when you are planning to visit but we found out the hard way that St Peter's is closed for three weeks (starting last Saturday) due to the congress/meeting that is taking place. Very disappointing.

Luann, I'm not seeing any closure of the basilica aside from the Wed. morning Papal general audiences and special masses on three Sundays this month. The Vatican website is showing some open dates for reserving combo tours of the museums/basilica, and I'm seeing skip-the-line tour tickets being sold for the basilica this month on at least one other site as well. Walks of Italy is also showing openings for their Pristine Sistine tour (which includes the basilica) this month.

Is it possible that you tried to go on Sunday 10/4 (closed for all or part of the day for opening of the Synod of Bishops) and misunderstood that closures were only for three Sundays (4th, 18th and 28th) and not for three entire WEEKS?

I would think that a closure that long would be very big news in the travel community but yours is the only mention I've seen.

I was wondering if a ticket is necessary to attend the 17:00 mass at St. Peter's Basilica.

Not unless it's a special Papal mass.

The weekday 17:00 mass is in Latin.
On Sunday, mass is at 17:45 and is in Italian.

http://www.vatican.va/various/basiliche/san_pietro/it/vita_liturgica/orari.htm

The basilica is always free for visiting or for attending mass unless otherwise scheduled. You do have to go through a security check, however, and be appropriately dressed. Tickets required for Papal services are also free; you just have to apply for them in advance.

Thank you, Kathy. I didn't know that all I had to do was to fall in line to get inside the basilica to hear mass.

The lovely thing about Italian churches is that almost all of them are free for visiting and worship/reflection. In fact, St Pete's will be happy to have someone coming for a service and not just to sightsee the basilica!

FYI. You don't need a printed reservation from the Vatican site. You can just show the email you received on your phone or tablet and the QR code that is in the email is scanned. No problem.