Islamic weights and measures and their modern equivalents
Question

                What are the equivalents of the mudd, saa‘, bareed, dhiraa‘, farsakh, and qullah in modern terms?
Praise be to Allah.Firstly:
A number of modern researchers have studied the weights, measures and lengths mentioned in the Qur’an and Sunnah. The scholars have discussed guidelines on their values in the past, and mentioned their equivalents in modern terms. These scholars include the following:

Mahmoud Faakhoori and Salaah ad-Deen Khawwaam, in their book Mawsoo‘ah Wahdaat al-Qiyaas al-‘Arabiyyah wa’l-Islamiyyah wa maa yu‘aadiluhaa bi’l-Maqaadeer al-Hadeethah: al-Atwaal, al-Masaahaat, al-Awzaan, al-Makaayeel.
Muhammad Najm ad-Deen al-Kurdi, in his book al-Maqaadeer ash-Shar‘iyyah wa’l-Ahkaam al-Muta‘alliqah biha – Kayl, Wazn, Miqyaas – mundhu ‘Ahd an-Nabi (sall-Allahu ‘alayhi wa sallam) wa Taqweemuha bi’l-Mu‘aasir.
‘Ali Pasha Mubaarak, in his book al-Mizaan fi’l-Aqyisah wa’l-Awzaan.
Muneer Hammood Farhaan al-Kubaysi, in his book al-Maqaadeer ash-Shar‘iyyah wa Ahammiyatuha fi Tatbeeq ash-Sharee‘ah al-Islamiyyah: al-Awzaan wa’l-Makaayeel wa’l-Maqaayees.
Muhammad al-Jaleeli, in his book al-Makaayeel wa’l-Awzaan wa’n-Nuqood al-‘Arabiyyah.
Muhammad Subhi al-Hallaaq, in his book al-Eedaahaat al-‘Asriyyah li’l-Maqaayees wa’l-Makaayeel wa’l-Awzaan ash-Shar‘iyyah.
Shaykh ‘Abdullah ibn Sulaymaan al-Munayyi‘, in his article Tahweel al-Mawaazeen wa’l-Makaayeel ash-Shar‘iyyah ila al-Maqaadeer al-Mu‘aasirah, published in Majallat al-Buhooth al-Islamiyyah (issue no. 59).
Shaykh Wahbah az-Zuhayli, in his book al-Fiqh al-Islami wa Adillatuhu.

Secondly:
With regard to what the questioner wants to know about the modern equivalents of those measures, weights and volumes, we say – with the addition of the equivalent of the “meel”:

Mudd

Shaykh ‘Abdullah ibn Sulaymaan al-Munayyi‘ (may Allah preserve him) said:
The mudd is equivalent to 544 grams, based on the fact that the mudd is equal to one and a third ratls, and the ratl is equivalent to 408 grams, as noted above.
Majallat al-Buhooth al-Islamiyyah (59/179).

Saa‘

Shaykh ‘Abdullah ibn Sulaymaan al-Munayyi‘ (may Allah preserve him) said:
The Council of Senior Scholars in the Kingdom of Saudi Arabia researched the equivalent of the saa‘ in kilograms, basing their research on the fact that the saa‘ of the Messenger of Allah (blessings and peace of Allah be upon him) was equal to four mudds, and the mudd was what could fill the cupped hands of a man of average size, then it was worked out  what was meant by “what could fill the cupped hands of a man of average size”. The conclusion reached was that the mudd is equivalent to approximately 650 grams, so the equivalent of the saa‘ is 2600 grams.
Majallat al-Buhooth al-Islamiyyah (59/178).

Qullah

Shaykh ‘Abdullah ibn Sulaymaan al-Munayyi‘ (may Allah preserve him) said:
The qullah that is well known among the scholars is the “qullah of Hajar”. This Hajar was a village near Madinah, not the Hajar of al-Bahrain.
We have mentioned above that one qullah was equal to two hundred and fifty ratls, and we have mentioned above that the ratl is equivalent to 408 grams. Therefore the equivalent of a single qullah in grams is 250 x 408 = 102,000 grams, or 102 kilograms.
Majallat al-Buhooth al-Islamiyyah (59/184).

Dhiraa‘

Shaykh Wahbah az-Zuhayli (may Allah preserve him) said:
The dhiraa‘ [cubit] referred to in fiqh is the Haashimi dhiraa‘, which is equivalent to 61.2 centimetres.

Meel

The meel [mile] is 4,000 dhiraa‘s [cubits], i.e., 1,848 metres.

Farsakh

The farsakh [parasang] is 3 meels, or 5,544 metres, or 12,000 khutwah [steps].

Bareed

The Arabian bareed is equal to 4 farsakhs, or 22,176 metres, or 22.176 kilometres.
Al-Fiqh al-Islami wa Adillatuhu (1/74, 75).
And Allah knows best.