Short answer
The Quran calls it the tabut, a chest that angels brought back to the Children of Israel as a sign that Talut (Saul) was their rightful king. It held sakinah (tranquillity) from Allah and a remnant of what the families of Musa and Harun left behind. The Quran does not say where it is today.
Muslims believe what the Quran states: the tabut was real, it held tranquillity from Allah and relics of the families of Musa and Harun, and angels carried it. Details beyond that come from commentators' reports, many taken from Israelite sources, and are neither firmly affirmed nor denied.
The four schools agree on this.
Explanation
The story is in Surah al-Baqarah. After the time of Musa, the Children of Israel asked their prophet for a king to lead them in battle. Allah chose Talut, and when they objected that he was not rich or noble, their prophet told them that the sign of his kingship would be the return of the chest, "in which is assurance from your Lord and a remnant of what the family of Moses and the family of Aaron had left, carried by the angels" (2:248). Talut then led them out, and Dawud (David) killed Jalut (Goliath).
The word the Quran uses for what was in it is sakinah: calm and tranquillity sent down by Allah. The same word appears elsewhere. When a Companion was reciting the Quran and saw something like a cloud come down, the Prophet ﷺ said it was the sakinah that descended because of the Quran.
What else was inside? Commentators narrated from early authorities that it held pieces of the tablets given to Musa, his staff and some belongings of Harun. These reports are not from the Prophet ﷺ, so scholars mention them without making them part of belief. Stories about where the chest was hidden or will be found, including claims that the Mahdi will bring it out, rest on weak or unknown reports.
For Muslims the lesson is the one the Quran draws: Allah gives authority to whom He wills, and real tranquillity comes from Him. Read the stories of Musa and Dawud.
The evidence
Quran, Surah Al-Baqara 2:248
وَقَالَ لَهُمْ نَبِيُّهُمْ إِنَّ ءَايَةَ مُلْكِهِۦٓ أَن يَأْتِيَكُمُ ٱلتَّابُوتُ فِيهِ سَكِينَةٌۭ مِّن رَّبِّكُمْ وَبَقِيَّةٌۭ مِّمَّا تَرَكَ ءَالُ مُوسَىٰ وَءَالُ هَٰرُونَ تَحْمِلُهُ ٱلْمَلَٰٓئِكَةُ ۚ إِنَّ فِى ذَٰلِكَ لَءَايَةًۭ لَّكُمْ إِن كُنتُم مُّؤْمِنِينَ ﴿٢٤٨﴾
And their prophet said to them, "Indeed, a sign of his kingship is that the chest will come to you in which is assurance from your Lord and a remnant of what the family of Moses and the family of Aaron had left, carried by the angels. Indeed in that is a sign for you, if you are believers."
Describes the chest, its tranquillity, the relics of Musa and Harun and the angels carrying it.
Sahih al-Bukhari 4839
Sahih... تِلْكَ السَّكِينَةُ تَنَزَّلَتْ بِالْقُرْآنِ ...
"That was the tranquillity (sakinah) that came down because of the Quran."
Shows sakinah is a real thing Allah sends down, the same word used for what was in the chest.
Where the ruling is recorded
- Tafsir Ibn Kathir, Commentary on Surah al-Baqarah, the story of Talut. Explains the tabut and gathers the early reports about its contents.
- Ma'arif al-Quran (Mufti Muhammad Shafi), Commentary on Surah al-Baqarah, the story of Talut and Jalut. Explains the tabut and sakinah for readers in the Deobandi tradition.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.